Mathematics IA — Formula Sheet
All key formulas for TS Inter 1st Year Maths IA — Matrices, Vectors, Trigonometry, Inverse Trig, Hyperbolic Functions and Properties of Triangles. Based on official TGBIE Annual Plan 2025-26.
Ch. 3 Matrices — Key Formulas
- 1Order of matrix: m × n (m rows, n columns)
- 2Transpose: (Aᵀ)ᵀ = A; (AB)ᵀ = BᵀAᵀ; (A+B)ᵀ = Aᵀ + Bᵀ
- 3Determinant of 2×2: |A| = ad − bc for A = [[a,b],[c,d]]
- 4Adjoint of 2×2: adj A = [[d, −b],[−c, a]]
- 5Inverse: A⁻¹ = (1/|A|) · adj A; valid only when |A| ≠ 0
- 6Rank of matrix: maximum order of non-zero minor
- 7Consistency: r(A) = r([A|B]) → consistent; r(A) < r([A|B]) → inconsistent
- 8Unique solution: r(A) = r([A|B]) = n (number of unknowns)
Ch. 4 & 5 Vectors — Key Formulas
- 1Magnitude: |a⃗| = √(a₁² + a₂² + a₃²)
- 2Unit vector: â = a⃗ / |a⃗|
- 3Position vector of midpoint of AB: (a⃗ + b⃗) / 2
- 4Section formula (internal): P = (mb⃗ + na⃗) / (m+n)
- 5Dot product: a⃗·b⃗ = |a⃗||b⃗|cosθ = a₁b₁ + a₂b₂ + a₃b₃
- 6Angle between vectors: cosθ = (a⃗·b⃗) / (|a⃗||b⃗|)
- 7Perpendicular condition: a⃗·b⃗ = 0
- 8Cross product: |a⃗×b⃗| = |a⃗||b⃗|sinθ
- 9Cross product (component form): a⃗×b⃗ = (a₂b₃−a₃b₂)i − (a₁b₃−a₃b₁)j + (a₁b₂−a₂b₁)k
- 10Scalar triple product: [a⃗ b⃗ c⃗] = a⃗·(b⃗×c⃗)
- 11Area of triangle: ½|a⃗×b⃗|
- 12Area of parallelogram: |a⃗×b⃗|
- 13Collinear: a⃗×b⃗ = 0⃗ (for position vectors) or [a⃗ b⃗ c⃗] = 0 (for three vectors)
Ch. 6 Trigonometric Ratios — Key Formulas
- 1sin(A+B) = sinA·cosB + cosA·sinB
- 2sin(A−B) = sinA·cosB − cosA·sinB
- 3cos(A+B) = cosA·cosB − sinA·sinB
- 4cos(A−B) = cosA·cosB + sinA·sinB
- 5tan(A+B) = (tanA + tanB) / (1 − tanA·tanB)
- 6sin2A = 2sinA·cosA = 2tanA/(1+tan²A)
- 7cos2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1 = (1−tan²A)/(1+tan²A)
- 8tan2A = 2tanA / (1 − tan²A)
- 9sin3A = 3sinA − 4sin³A
- 10cos3A = 4cos³A − 3cosA
- 11sinC + sinD = 2sin((C+D)/2)·cos((C−D)/2)
- 12sinC − sinD = 2cos((C+D)/2)·sin((C−D)/2)
- 13cosC + cosD = 2cos((C+D)/2)·cos((C−D)/2)
- 14cosC − cosD = −2sin((C+D)/2)·sin((C−D)/2)
- 152sinA·cosB = sin(A+B) + sin(A−B)
- 162cosA·cosB = cos(A−B) + cos(A+B)
- 172sinA·sinB = cos(A−B) − cos(A+B)
Ch. 8 Inverse Trigonometric Functions — Key Formulas
- 1sin⁻¹(sinθ) = θ for θ ∈ [−π/2, π/2]
- 2cos⁻¹(cosθ) = θ for θ ∈ [0, π]
- 3tan⁻¹(tanθ) = θ for θ ∈ (−π/2, π/2)
- 4sin⁻¹(x) + cos⁻¹(x) = π/2
- 5tan⁻¹(x) + cot⁻¹(x) = π/2
- 6tan⁻¹(x) + tan⁻¹(y) = tan⁻¹((x+y)/(1−xy)) when xy < 1
- 72tan⁻¹(x) = sin⁻¹(2x/(1+x²)) = cos⁻¹((1−x²)/(1+x²)) = tan⁻¹(2x/(1−x²))
Ch. 9 Hyperbolic Functions — Key Formulas
- 1sinhx = (eˣ − e⁻ˣ)/2; coshx = (eˣ + e⁻ˣ)/2
- 2tanhx = sinhx/coshx = (eˣ − e⁻ˣ)/(eˣ + e⁻ˣ)
- 3cosh²x − sinh²x = 1
- 4sinh(x+y) = sinhx·coshy + coshx·sinhy
- 5cosh(x+y) = coshx·coshy + sinhx·sinhy
- 6sinh⁻¹(x) = ln(x + √(x²+1))
- 7cosh⁻¹(x) = ln(x + √(x²−1)), x ≥ 1
- 8tanh⁻¹(x) = ½ ln((1+x)/(1−x)), |x| < 1
Ch. 10 Properties of Triangles — Key Formulas
- 1Sine rule: a/sinA = b/sinB = c/sinC = 2R
- 2Cosine rule: cosA = (b²+c²−a²)/(2bc); similar for B, C
- 3Tangent rule (Napier's): tan((B−C)/2) = (b−c)/(b+c) · cot(A/2)
- 4Half-angle: sin(A/2) = √((s−b)(s−c)/bc); cos(A/2) = √(s(s−a)/bc)
- 5tan(A/2) = √((s−b)(s−c)/(s(s−a)))
- 6Area of triangle: Δ = ½ab·sinC = √(s(s−a)(s−b)(s−c)) [Heron's formula]
- 7In-radius: r = Δ/s
- 8Circum-radius: R = abc/(4Δ)
- 9Ex-radius: r₁ = Δ/(s−a); r₂ = Δ/(s−b); r₃ = Δ/(s−c)
- 10s = (a+b+c)/2 (semi-perimeter)